3 Answers
3

The sum is well-known (e.g., its decimal expansion is OEIS A153386), but
it seems(?) that although it is known to be irrational, and several special sums
of Fibonacci numbers are known to be transcendental, the Reciprocal Fibonacci constant itself (as it is known)
remains unsettled.

Let $\lbrace F_n\rbrace _{n\ge0}$ and $\lbrace L_n\rbrace _{n\ge0}$ be Fibonacci and Lucas numbers,
respectively,
$F_0=0$, $F_1=1$, $F_{n+2}=F_n+F_{n+1}$ for $n\ge0$, and
$L_0=2$, $L_1=1$, $L_{n+2}=L_n+L_{n+1}$ for $n\ge0$.
Using Nesterenko's theorem
[Yu.V. Nesterenko, Sb. Math.187:9 (1996), 1319--1348. MR1422383]
and expressing the series
$$
\zeta_F(2s)=\sum_{n=1}^\infty\frac1{F_n^{2s}}
\quad\text{and}\quad
\zeta_L(2s)=\sum_{n=1}^\infty\frac1{L_n^{2s}},
\qquad s=1,2,\dots,
\qquad (1)
$$
via the Eisenstein series
$$
E_{2s}(q)=1-\frac{4s}{B_{2s}}\sum_{n=1}^\infty\sigma_{2s-1}(n)q^n,
\qquad \sigma_k(n)=\sum_{d\mid n}d^k,
$$
where $B_{2s}\in\mathbb Q$ are Bernoulli numbers,
the authors prove the algebraic independence of
the numbers in the collections $\zeta_F(2)$, $\zeta_F(4)$, $\zeta_F(6)$
and $\zeta_L(2)$, $\zeta_L(4)$, $\zeta_L(6)$ as well as express
algebraically even "zeta values" $\zeta_F(2s)$ (and $\zeta_L(2s)$)
for $s\ge4$ in terms of the three algebraically independent numbers
in the corresponding collection. Similar algebraic independence results
are shown for the alternating versions of (1). Known irrationality
results for $\zeta_F(k)$ and $\zeta_L(k)$ with odd $k$
(when the series have no known relations with the modular world) are indicated.
It is worth mentioning that these results go in a natural parallel
with the ones for the so-called $q$-zeta values defined in
[W. Zudilin, Math. Notes72:5-6 (2002), 858--862. MR1964151]
and [C. Krattenthaler, T. Rivoal, and W. Zudilin,
J. Inst. Math. Jussieu5:1 (2006), 53--79. MR2195945].
In particular, it is natural to expect a "Fibonacci" analogue of Rivoal's theorem
[T. Rivoal, C. R. Acad. Sci. Paris Ser. I Math.331:4 (2000), 267--270. MR1787183]
on the infiniteness of
irrational numbers in the set $\zeta_F(1),\zeta_F(3),\zeta_F(5),\dots$
(or $\zeta_L(1),\zeta_L(3),\zeta_L(5),\dots$),
based on the techniques developed in the paper under review and
in the joint paper of Krattenthaler, Rivoal and the reviewer cited above.

Reviewed by Wadim Zudilin

To summarize, the difficulty of proving the transcendence for odd $\zeta_F(s)$ with $s$ odd is similar to the one for odd zeta values. The irrationality of $\zeta_F(1)$ is known but already its non-quadraticity remains an open problem.

This looks kind of like the logarithmic derivative of the modular form $\prod (1-q^n)$, but it's not exactly that. In any case, it is unlikely that someone has evaluated this sort-of-a-modular-form at this particular quadratic irrational. The natural thinkg to do is to evaluate modular forms when $\tau$ is a quadratic irrational, where $q=e^{2 \pi i \tau}$. For example, people might know what this sum equals at $q=e^{- \pi \sqrt{163}}$.

I'd be pleasantly surprised if there is a known answer. Of course, the safe money is always to bet on "transcendental" when you don't see a reason to expect anything else.

Looks like I was too pessimistic. It is known to be irrational, see http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html (thanks Qiaochu!). They do indeed describe this quantity in terms of theta functions, which are a type of modular forms. To my surprise, they are able to prove things about these modular forms at $q$, although not to answer this question. Anyway, it looks like this reference has as much information as you can hope for.